GPS-based in-vehicle sensor calibration algorithm

ABSTRACT

A system and method for calibrating the bias and scale factors of a heading rate sensor, such as a yaw-rate sensor, using GPS signals. The system receives wheel speed or rotation signals, a vehicle odometer reading, GPS signals and yaw-rate signals. The system includes a wheel-slip detection processor that determines whether there is wheel-slip based on the wheel speed signals and the GPS signals. The system also includes a wheel-based acceleration processor that estimates vehicle acceleration. The system also includes a differential odometry processor that determines vehicle heading based on wheel speed. The system also includes a GPS reference data validation processor that determines whether the GPS signals are valid using the estimated vehicle acceleration and wheel speeds. The valid GPS signals are then used to calibrate the yaw-rate sensor signals, which can be used for vehicle heading purposes.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to a system and method for calibratinga heading rate sensor, such as an inertial sensor, i.e., a yaw-ratesensor, and, more particularly, to a system and method for removingsensor bias and scale factor errors from a heading rate sensor to usethe sensor signals to provide an accurate vehicle heading.

2. Discussion of the Related Art

GPS signals, or other Global Navigation Satellite System (GNSS) signals,can provide accurate positioning and navigation. However, GPS receiverssuffer from sky visibility-related limitations, for example, in urbancanyons and areas with dense tree cover. Further, GPS signals may sufferfrom related multi-path errors or cross-correlation errors in suchareas. Because of existing highly sensitive and fast reacquisition GPStechnology, accurate GPS signals become available when sky visibility istemporarily improved for short durations, such as 10-20 seconds, even inless than optimum environments. Therefore, the continuity of GPStechnology comes down to maintaining positioning accuracy through GPSoutages between GPS available time windows.

Automotive-grade inertial sensors, such as yaw-rate sensors andaccelerometers, have highly variable bias and scale characteristics thatcause sensor drift that typically makes them un-suitable for navigationand heading determination functions without proper error correctiontechniques. For example, certain automotive-grade yaw-rate sensors allowup to 2 deg/sec variations for the yaw-rate sensor bias. If such avariability is not corrected, and is allowed for over a period of twominutes, a yaw-rate sensor starting with a bias of 0 deg/sec at zeroseconds could reach a bias of 2 deg/sec after 120 seconds. If a lineargrowth of bias were assumed for simplicity, a heading change derived byintegrating yaw-rate sensor signals that is not calibrated wouldindicate a heading change of 120° only as a result of the variation ofthe bias.

Inertial sensors can be used in combination with GPS receivers toprovide a reasonably accurate vehicle heading, and position if adistance measure, such as vehicle wheel speeds, are available, even whenthe GPS signals are not available. However, automotive-grade inertialsensors do not typically provide the same level of accuracy as GPSsignals. GPS/inertial sensor integrated systems can calibrate theinertial sensors and maintain vehicle heading and position accuracyusing GPS signals when the GPS signals are available, and use thecalibrated inertial sensors when the GPS signals are not available tomaintain a heading and a position solution until the GPS signals becomeavailable again.

Known yaw-rate sensor calibration algorithms typically approach bias andscale calibration as a two-step process, and require specific vehiclemaneuvers to be performed for the calibration. For example, sensor biascalibration may require the vehicle to be driven in a straight line orbe stationary for a known period of time so that the accumulated yawheading error can be directly estimated as a result of sensor biaserror. For scale calibration, the vehicle may be required to be driventhrough a controlled turn to provide scale calibration.

SUMMARY OF THE INVENTION

In accordance with the teachings of the present invention, a system andmethod are disclosed for calibrating the bias and scale factors of aheading rate sensor, such as a yaw-rate sensor, using GPS signals. Thesystem receives wheel speed or rotation signals, a vehicle odometerreading, GPS signals and yaw-rate signals. The system includes awheel-slip detection processor that determines whether there iswheel-slip based on the wheel speed signals and the GPS signals. Thesystem also includes a wheel-based acceleration processor that estimatesvehicle acceleration. The system also includes a differential odometryprocessor that determines vehicle heading based on wheel speed. Thesystem also includes a GPS reference data validation processor thatdetermines whether the GPS signals are valid using the wheel speed andthe estimated vehicle acceleration. The valid GPS signals are then usedto calibrate the yaw-rate sensor signals, which can be used for vehicleheading purposes.

Additional features of the present invention will become apparent fromthe following description and appended claims taken in conjunction withthe accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graph with time on the horizontal axis and heading on thevertical axis including graph lines showing vehicle heading provided byGPS signals, uncalibrated yaw-rate sensor signals, bias calibratedyaw-rate sensor signals, and bias and scale calibrated yaw-rate sensorsignals;

FIG. 2 is a plan view of a vehicle including a system for providinginertial sensor bias and scale calibration, according to an embodimentof the present invention; and

FIG. 3 is a block diagram of the bias and scale calibration system shownin FIG. 2.

DETAILED DESCRIPTION OF THE EMBODIMENTS

The following discussion of the embodiments of the invention directed toa system and method for providing inertial sensor bias and scalecalibration to provide accurate vehicle heading readings is merelyexemplary in nature, and is in no way intended to limit the invention orits applications or uses.

The present invention proposes a system and method that uses GPS signalsto calibrate the scale and bias of a heading rate sensor, such as ayaw-rate sensor, to maintain the accuracy of the inertial sensor so asto allow the use of automotive-grade yaw-rate sensors for navigation,positioning, heading and enhanced vehicle stability control functionswhen the GPS signals are unavailable. The algorithm of the inventioncalibrates both bias and scale factor using the same set of data, anddoes not require specific vehicle maneuvers to be performed to achievecalibration. In one non-limiting embodiment, a yaw-rate and GPS headingdata set at a 1 Hz or greater data rate that was 40 seconds long wasused for the calibration of a yaw-rate sensor, during which a region ofrelative straight driving and one or more vehicle turns areidentifiable.

FIG. 1 is a graph with time on the horizontal axis and vehicle headingon the vertical axis that includes a graph line 10 showing vehicleheading from GPS signals and a graph line 12 showing vehicle heading fora yaw-rate sensor that has not been calibrated for bias and scale. Graphline 14 shows vehicle heading from a yaw-rate sensor that has been biascalibrated by the algorithm of the invention, and graph line 16 showsvehicle heading from a yaw-rate sensor that has been bias and scalecalibrated by the algorithm of the invention.

FIG. 2 is a plan view of a vehicle 20 including front wheels 22 and 24and rear wheels 26 and 28. The vehicle 20 also includes a bias and scalecalibration system 30, according to an embodiment of the presentinvention. The wheels 22, 24, 26 and 28 each include a wheel speedsensor 32, 34, 36 and 38, respectively, that provide wheel speed and/orwheel rotation signals to the system 30. A GPS receiver 42 provides GPSsignals to the system 30 and an odometer 44 provides vehicle odometersignals, particularly a drive shaft count, to the system 30.Additionally, the system 30 receives vehicle yaw-rate signals from ayaw-rate sensor 46 and vehicle lateral acceleration signals from alateral acceleration sensor 48.

FIG. 3 is a block diagram of the system 30, where the wheel speed and/orwheel rotation signals from the sensors 32, 34, 36 and 38 are providedon line 52, the odometer signals from the odometer 44 are provided online 54, the GPS signals from the GPS receiver 42 are provided on line56 and the yaw-rate signals from the yaw-rate sensor 46 are provided online 58. As will be discussed in further detail below, the system 30provides a vehicle heading estimation that can be used for any suitablepurpose, such as a digital compass, vehicle navigation, vehiclestability control, etc. When the GPS signals are available, the system30 uses those signals to provide the vehicle heading and uses the GPSsignals to calibrate the bias and scale factors of the yaw-rate sensor46. When the GPS signals are not available, the system 30 uses thepreviously calibrated yaw-rate signals to provide the vehicle heading.When the GPS signals are not available and the yaw-rate sensor 46 is notoperating properly, then the system 30 uses wheel speed signals toestimate the vehicle heading rate using differential odometry that hasbeen bias and scale calibrated using the calibration algorithm of theinvention in the same way that the yaw-rate sensor is calibrated toprovide the vehicle heading. Although it is the yaw-rate sensor 46 thatis being calibrated by the system 30 for vehicle heading purposes,alternately the algorithm of the present invention for calibration canbe used for any vehicle heading rate sensor, such as a differentialodemetry sensor.

The validation of the accuracy of the GPS reference data is one part ofthe sensor calibration provided by the invention. For the invention,vehicle velocity and position estimates are considered the available GPSreference velocity data for the calibration process. Low-cost GPSreceivers typically use pseudo-ranges or carrier-smooth pseudo-rangesfor position estimation and Doppler effect or pseudo-range rateobservations for velocity estimation. Out of all of the GPS measurementsmentioned, Doppler effect observations are affected the least bymulti-path errors. More importantly, multi-path errors are also thedominant error contributor for automotive navigation and positioningapplications. Therefore, Doppler effect derived GPS velocities areconsiderably better in terms of accuracy and reliability, andavailability compared to GPS position estimates.

The algorithm of the invention uses the ratio between GPS reportedvehicle velocity and wheel rotator/speed to verify the accuracy of theGPS reference data. This enables a much more reliable verificationwithout using position domain data. This verification is performed inaddition to the generic GPS data validation techniques based on thenumber of visible GPS satellites used by the GPS receiver 42 to generatea solution and signal-to-noise ratio, satellite elevation and positionestimation least square residuals of each satellite.

The system 30 includes a wheel-slip detection processor 60 that uses thewheel speed and/or wheel rotation signals from each of the sensors 32,34, 36 and 38 on the line 52 and the GPS signals on the line 56 todetect wheel-slip and minimize the corruption of the GPS validationmechanism as a result of any of the wheels 22-28 slipping. The processor60 serves as the primary data verification process, and uses a simplewheel velocity-to-odometer velocity ratio to determine if the wheelcounts include errors as a result of wheel-slip. According to onenon-limiting embodiment, the processor 60 employs a model from analgorithm based on equation (1) below to determine whether wheel-slip ofany of the wheels 22-28 exists.

$\begin{matrix}{{\frac{v_{i,{GPS}}}{v_{i,{sensor}}}} \leq \delta} & (1)\end{matrix}$

Where ν_(GPS) is GPS-based velocity, ν_(sensor) is wheel sensor basedvehicle velocity and δ is a predetermined data quality threshold.

If the processor 60 determines that there is no wheel-slip, then thewheel speed and/or rotation signals and the odometer signals are sent toor can properly be used by a wheel-based acceleration estimationprocessor 62 that estimates vehicle acceleration. The estimationprocessor 62 uses a simple time differentiation of wheel velocity toestimate the vehicle acceleration. GPS position and velocity datastreams often lag other in-vehicle data streams due to processingdelays. This is especially true if filtering is implemented as a part ofthe GPS signal processing and estimation. Most alternative sensorenabled GPS receivers, i.e., where vehicle sensor data is used for GPSposition and velocity estimation and aiding, send out trigger pulses toread vehicle data messages and subsequently combine the data with theGPS signals to provide an internally combined solution. Therefore,vehicle speed at a particular time is available through the wheel speedsensors 32, 34, 36 and 38 before the corresponding GPS speed estimate isavailable. Although the magnitude of this lag may vary, existinghardware and software has shown tens of milliseconds to a second ofdelay, the worst case being the alternative sensor enabled GPSreceivers.

The processor 62 uses the estimated vehicle acceleration derived fromthe wheel speed sensors 32, 34, 36 and 38 to identify the speed ratio(GPS speed to wheel-based speed) variation resulting from the time lagof the GPS signals because they are highly correlated. Thus, ratiovariations corresponding to GPS time lag during acceleration anddeceleration are not misidentified as corrupted GPS reference data.

The GPS signals from the GPS receiver 42 on the line 56 and the wheelspeed/rotation signals from the processor 60 are sent to a resamplingand time synchronization processor 66 to synchronize the time framebetween the GPS signals and the wheel speed/odometer readings so thatthe wheel sensor data stream is synchronized with the GPS data stream.Typically, vehicle sensor data is available at a higher sampling ratethan GPS signals, and therefore, the GPS signal rate governs the rate ofdata sent to a GPS validation process. For example, GPS signals may beat 1 Hz and vehicle sensor data may be at 10 Hz or higher.

The time synchronized GPS signals and wheel speed/rotation signals fromthe processor 66 are sent to a GPS reference data validation processor68 along with the acceleration estimation signals from the processor 62.The processor 68 selects valid GPS reference data that fulfills certainrequirements for sensor calibration. The output of the processor 68 is asignal identifying whether the GPS signals for a particular period oftime are valid or not. In one non-limiting embodiment, the algorithmuses a model based on equation (2) below to provide the validation ofthe GPS reference data in the processor 68.

$\begin{matrix}{{\frac{v_{i,{GPS}}}{v_{i,{sensor}}}} \leq {ka}_{i}} & (2)\end{matrix}$

Where k is a predefined constant depending on the GPS receiver used, andα_(i) is the vehicle acceleration estimated by wheel data timedifferencing.

The yaw-rate sensor signals on the line 58 and the valid GPS signalsfrom the processor 68 are sent to a bias and scale calibration processor70. The processor 70 uses the valid GPS signals to remove the bias andcalibrate the scale factor of the yaw-rate sensor 46 so that theyaw-rate sensor 46 can be used for vehicle heading purposes when the GPSsignals are not valid. In one embodiment, the calibration algorithm usedin the processor 70 treats the GPS heading profile and the yaw-rateheading profile as two shapes, i.e., generated by time integrating theyaw-rate signal, and attempts to estimate the bias and scale factors sothat a minimum is reached in the model. This process is shown byequation (3) below.

$\begin{matrix}{\min\limits_{S,B}{\sum\limits_{i - 1}^{N}{\alpha_{i}^{2}( {{GPS}_{i} - {S \times ( {{Sensor}_{i} - B} )}} )}}} & (3)\end{matrix}$

Where N is the number of data points, α_(i) is the relative importanceof epoch i, GPS_(i) is the GPS heading (time=i), S is the scale factor,B is yaw bias and Sensor_(i) is the yaw-based heading derived, forexample, using Equation (6) below (time=i).

The model given by equation (3) also allows the inclusion of relativeimportance weights for individual observations in the calibration dataset. For example, if several erroneous heading observations are detectedin the GPS heading dataset, i.e., detected using continuity of GPSheading data, vehicle dynamic constraints and yaw-rate based heading,which has a much greater continuity, those data points can be assignedless weights or even be ignored in the parameter optimization process.

Prior to optimizing the model of equation (3), the algorithm establishestwo search spaces for the bias and scale parameters. Based on theheading discrepancy between the GPS readings and yaw-rate based headingestimation using the perfect yaw-rate sensor assumption, the algorithmestimates a bias value and defines a search space around thisapproximated bias. It also picks identifiable vehicle turns, such as byusing uncalibrated yaw-rate signals, and estimates an approximate scalefactor along with an error estimate resulting in a search space for thescale factor. Subsequently, a search is performed to estimate theoptimum bias and scale factor values.

The actual implementation of the calibration algorithm may varydepending on the integration mechanism used for GPS signals and vehiclesensors. For example, vehicle heading derived from calibrated sensorsmay be fed back to a GPS position and velocity estimation processor in acomplex implementation. In any implementation, frequent calibration ofthe sensor will improve the accuracy of the estimated heading. Ideally,the calibration will take place as a continuous process using the mostrecent segment of valid GPS data. However, this may be not feasiblebecause of limited processing resources in a vehicle platform, thusrequiring an automated scheme to trigger a new calibration updatewhenever valid reference data is available and when significant sensorbias or scale deviations are detected.

A model based on equations (4) and (5) below can be used to monitorsignificant variations in the heading reported by the validated GPS datasegments and corresponding calibrated sensor data to trigger an updateof the calibration parameters.

$\begin{matrix}{{\max {\frac{\varphi_{i,{Sensor}}}{\varphi_{i,{GPS}}}}} < \delta_{Scale}} & (4) \\{{\max {{\phi_{i,{Sensor}} - \varphi_{i,{GPS}}}}} < \delta_{Bias}} & (5)\end{matrix}$

Where φ_(Sensor) is the sensor-based heading (yaw-rate or differentialodometry), φ_(GPS) is the GPS-based heading, δ_(Scale) is the scalefactor threshold and δ_(Bias) is the bias threshold.

The wheel speed and/or rotation signals and the odometer signals arealso sent to or can be used by a differential odometry processor 64. Thedifferential odometry processor 64 uses the wheel rotation counts fromthe wheel speed sensors 32, 34, 36 and 38 to determine the vehicleheading based on the distance between two of the wheels, such as wheels22 and 26 or 24 and 28. The bias and scale calibration algorithm in theprocessor 70 can also be used to calibrate the bias and scale of thedifferential odometry signals.

The bias and scale calibration factors from the processor 70 are thensent to a heading estimation processor 72 along with the yaw-ratesignals on the line 58. The heading estimation algorithm used in theprocessor 72 estimates the vehicle heading using the yaw-rate signalsthat have been calibrated by the algorithm in the processor 70. In onenon-limiting embodiment, the scale and bias factors can be used toestimate the calibrated yaw-rate sensor-based heading by using the modelin equation (6) below.

φ_(i)=φ_(i−1) +S(∂φ_(i)−Γ)dT   (6)

Where φ is the yaw-rate sensor-based heading, ∂φ is the yaw-rate, S isthe scale factor, Γ is the bias factor and dT is the yaw rate sensordata interval (1/sampling rate).

If the yaw-rate signals are not available at any particular time, thenthe heading estimation processor 72 can use the heading signals from theodometry processor 64. Various things would cause the processor 72 notto use the yaw-rate sensor signals on the line 58, such as sensorfailure. It is well known in the art to provide vehicle heading by thewheel speed of two front and rear wheels of a vehicle. However, as iswell understood in the art, it is less accurate than providing vehicleheading using yaw-rate sensors.

The heading estimation from the processor 72 can then be used in anysuitable system on the vehicle 20, such as a digital compass 74.

The foregoing discussion discloses and describes merely exemplaryembodiments of the present invention. One skilled in the art willreadily recognize from such discussion and from the accompanyingdrawings and claims that various changes, modifications and variationscan be made therein without departing from the spirit and scope of theinvention as defined in the following claims.

1. A system for determining the heading of a vehicle, said systemcomprising: a GPS receiver providing GPS signals indicating the positionof the vehicle; a yaw-rate sensor for providing yaw-rate signalsindicating the yaw-rate of the vehicle; an acceleration estimationprocessor for determining the acceleration of the vehicle and providingacceleration signals; a GPS validation processor receiving the GPSsignals, wheel speed or rotation signals and the acceleration signals,and determining whether the GPS signals are valid; and a bias and scalecalibration processor responsive to the yaw-rate sensor signals and theGPS signals, said calibration processor providing bias and scalecalibration factors for the yaw-rate sensor signals using the GPSsignals if the validation processor determines that the GPS signals arevalid.
 2. The system according to claim 1 further comprising a wheelspeed sensor for providing the wheel speed or rotation signals of thespeed or rotation of wheels on the vehicle and a wheel-slip detectionprocessor responsive to the wheel speed signals, said wheel-slipdetection processor determining whether wheel-slip is present.
 3. Thesystem according to claim 2 wherein the wheel-slip detection processoruses the following equation to determine wheel-slip:${\frac{v_{i,{GPS}}}{v_{i,{sensor}}}} \leq \delta$ where ν_(GPS) isGPS-based velocity, ν_(sensor) is wheel sensor based vehicle velocityand δ is a predetermined data quality threshold.
 4. The system accordingto claim 1 further comprising a heading estimation processor responsiveto the bias and scale calibration factors from the calibration processorand the yaw-rate signals, said heading estimation processor using thecalibration factors and the yaw-rate signals to provide vehicle heading.5. The system according to claim 4 wherein the heading estimationprocessor uses the following equation to provide the vehicle heading:φ_(i)=φ_(i−1) +S(∂φ_(i)−Γ)dT where φ is the yaw-rate sensor-basedheading, ∂φ is the yaw-rate, S is the scale factor, Γ is the bias factorand dT is a yaw rate sensor data interval.
 6. The system according toclaim 1 wherein the GPS validation processor uses a ratio betweenvelocity provided by the GPS signal s and wheel speed or rotationsignals to determine whether the GPS signals are valid.
 7. The systemaccording to claim 1 wherein the acceleration estimation processor usestime differentiation of wheel velocity to estimate the vehicleacceleration.
 8. The system according to claim 1 wherein the bias andscale calibration processor uses the following equation to determine thebias and scale calibration factors:$\min\limits_{S,B}{\sum\limits_{i - 1}^{N}{\alpha_{i}^{2}( {{GPS}_{i} - {S \times ( {{Sensor}_{i} - B} )}} )}}$where N is the number of data points, α_(i) is the relative importanceof epoch i, GPS_(i) is the GPS heading (time=i), S is the scale factor,B is yaw bias and Sensor_(i) is the yaw-based heading.
 9. The systemaccording to claim 1 further comprising a resampling and timesynchronization processor that synchronizes the time frame between theGPS signals and wheel speed signals.
 10. The system according to claim 1further comprising a differential odometry processor for determiningvehicle heading based on wheel rotation counts.
 11. A system forcalibrating bias and scale of a sensor in a vehicle, said systemcomprising: a heading rate sensor providing sensor signals; a GPSreceiver providing GPS signals indicating the position of the vehicle; aplurality of wheel speed/wheel rotation sensors for providing signals ofthe wheel speed or wheel rotation of wheels on the vehicle; a wheel-slipdetection processor responsive to the wheel speed or wheel rotationsignals and determining whether any of the wheels have wheel-slip; anacceleration estimation processor for determining the acceleration ofthe vehicle based on the wheel speed or wheel rotation signals; and abias and scale calibration processor responsive to the sensor signalsand the GPS signals, said calibration processor providing bias and scalecalibration factors for the sensor signals using the GPS signals. 12.The system according to claim 11 further comprising a heading estimationprocessor responsive to the bias and scale calibration factors from thecalibration processor and the yaw-rate signals, said heading estimationprocessor using the calibration factors and the yaw-rate signals toprovide vehicle heading.
 13. The system according to claim 11 whereinthe acceleration estimation processor uses time differentiation of wheelvelocity to estimate the vehicle acceleration.
 14. A system fordetermining the heading of a vehicle, said system comprising: a GPSreceiver providing GPS signals indicating the position of the vehicle; ayaw-rate sensor for providing yaw-rate signals indicating the yaw-rateof the vehicle; a plurality of wheel speed/wheel rotation sensors forproviding signals of the wheel speed or wheel rotation of wheels on thevehicle; a wheel-slip detection processor responsive to the wheel speedor wheel rotation signals and determining whether any of the wheels havewheel-slip; an acceleration estimation processor for determining theacceleration of the vehicle based on the wheel speed or wheel rotationsignals; a GPS validation processor receiving the GPS signals and theacceleration signals, and determining whether the GPS signals are valid;a bias and scale calibration processor responsive to the yaw-rate sensorsignals and the GPS signals, said calibration processor providing biasand scale calibration factors for the yaw-rate sensor signals using theGPS signals if the validation processor determines that the GPS signalsare valid; and a heading estimation processor responsive to the bias andscale calibration factors from the calibration processor and theyaw-rate signals, said heading estimation processor using thecalibration factors and the yaw-rate signals to provide vehicle heading.15. The system according to claim 14 wherein the wheel-slip detectionprocessor uses the following equation to determine wheel-slip:${\frac{v_{i,{GPS}}}{v_{i,{sensor}}}} \leq \delta$ where ν_(GPS) isGPS-based velocity, ν_(sensor) is wheel sensor based vehicle velocityand δ is a predetermined data quality threshold.
 16. The systemaccording to claim 14 wherein the heading estimation processor uses thefollowing equation to provide the vehicle heading:φ_(i)=φ_(i−1) +S(∂φ_(i)−Γ)dT where φ is the yaw-rate sensor-basedheading, ∂φ is the yaw-rate, S is the scale factor, Γ is the bias factorand dT is a yaw rate sensor data interval.
 17. The system according toclaim 14 wherein the GPS validation processor uses a ratio betweenvelocity provided by the GPS signal and the wheel speed or rotationsignals to determine whether the GPS signals are valid.
 18. The systemaccording to claim 14 wherein the acceleration estimation processor usestime differentiation of wheel velocity to estimate the vehicleacceleration.
 19. The system according to claim 14 wherein the bias andscale calibration processor uses the following equation to determine thebias and scale calibration factors:$\min\limits_{S,B}{\sum\limits_{i - 1}^{N}{\alpha_{i}^{2}( {{GPS}_{i} - {S \times ( {{Sensor}_{i} - B} )}} )}}$where N is the number of data points, α_(i) is the relative importanceof epoch i, GPS_(i) is the GPS heading (time=i), S is the scale factor,B is yaw bias and Sensor_(i) is the yaw-based heading.
 20. The systemaccording to claim 14 further comprising a resampling and timesynchronization processor that synchronizes the time frame between theGPS signals and wheel speed signals.
 21. The system according to claim14 further comprising a differential odometry processor for determiningvehicle heading based on wheel rotation counts.